In contrast to point pattern based methods, we can view the location of cells or spots as a fixed lattice and measure the corresponding marker expression at each location. HTS-based spatial transcriptomic technologies often produce data on a regular lattice (i.e., approximately evenly spaced spots or beads of uniform size and shape), whereas imaging-based technologies yield irregular lattice structures (i.e., with variable cell sizes and shapes, and non-uniform spacing).
For this representation of the cells, we will rely on the SpatialFeatureExperiment package (see below for more details). For preprocessing of the dataset, we refer the reader to the vignette of the Voyager package (Moses et al. 2023). The Voyager package also provides wrapper functions around the package spdep (Pebesma and Bivand 2023) that work directly on the SpatialFeatureExperiment object. The package spdep is designed for the analysis of spatial data with lattice structure.
suppressPackageStartupMessages({
library(dplyr)
library(ggplot2)
library(patchwork)
library(Voyager)
library(SpatialFeatureExperiment)
library(SFEData)
library(spdep)
library(sf)
library(stringr)
library(tidyr)
library(magrittr)
library(scater)
})
theme_set(theme_light())We will load a dataset generated by (McKellar et al. 2021) using the Visium technology (Ståhl et al. 2016). The data shows a sample taken from the tibialis anterior muscle of a mouse.
# Load the dataset
sfe <- SFEData::McKellarMuscleData(dataset = "full")
# Take spots that are covered with tissue
sfe_tissue <- sfe[, colData(sfe)$in_tissue]
# Filter out genes with no counts
sfe_tissue <- sfe_tissue[rowSums(counts(sfe_tissue)) > 0, ]
# Convert counts log-transformed normalized expression values
sfe_tissue <- scater::logNormCounts(sfe_tissue)SpatialFeatureExperiment (Moses et al. 2023) objects are an extension of the SpatialExperiment object (Righelli et al. 2022). It additionally contains geometric annotations that are encoded as simple features of the sf library (Pebesma and Bivand 2023).
sfe_tissueclass: SpatialFeatureExperiment
dim: 15043 932
metadata(0):
assays(2): counts logcounts
rownames(15043): ENSMUSG00000025902 ENSMUSG00000096126 ...
ENSMUSG00000064368 ENSMUSG00000064370
rowData names(6): Ensembl symbol ... vars cv2
colnames(932): AAACATTTCCCGGATT AAACCTAAGCAGCCGG ... TTGTGTTTCCCGAAAG
TTGTTGTGTGTCAAGA
colData names(13): barcode col ... in_tissue sizeFactor
reducedDimNames(0):
mainExpName: NULL
altExpNames(0):
spatialCoords names(2) : imageX imageY
imgData names(1): sample_id
unit: full_res_image_pixels
Geometries:
colGeometries: spotPoly (POLYGON)
annotGeometries: tissueBoundary (POLYGON), myofiber_full (POLYGON), myofiber_simplified (POLYGON), nuclei (POLYGON), nuclei_centroid (POINT)
Graphs:
Vis5A: For example, the spots of the Visium dataset are stored as a simple feature collection.
colGeometry(sfe_tissue, "spotPoly") |> head()Simple feature collection with 6 features and 0 fields
Geometry type: POLYGON
Dimension: XY
Bounding box: xmin: 5857.695 ymin: 4126.151 xmax: 10206.53 ymax: 11789.24
CRS: NA
geometry
1 POLYGON ((6037.973 5430.826...
2 POLYGON ((7977.344 4461.467...
3 POLYGON ((8547.633 11699.1,...
4 POLYGON ((6867.944 4463.856...
5 POLYGON ((9779.598 4216.29,...
6 POLYGON ((10206.53 9282.577...Lattice data refers to spatial data collected at locations arranged in a regular or irregular grid (lattice). Each location has a defined spatial unit, and the sampling locations are fixed rather than random. This approach contrasts with point pattern analysis, where we assume that the locations were generated by a stochastic process (Zuur, Ieno, and Smith 2007; Pebesma and Bivand 2023).
As can be seen below, the Visium dataset is a regular lattice. The color shows the number of counts detected in each Visium spot. In contrast, the outlines of individual cells after segmentation (e.g., from a higher resolution spatial omics assay) could be seen as an irregular lattice. As this dataset also contains (manual) segmentations of myofibers (muscle cells stored as myofiber_simplified) we will illustrate calculations based on irregular lattice on the myofiber segmentations.
# A plot using the plotSpatialFeature from Voyager
p <- plotSpatialFeature(sfe_tissue,
"nCounts",
annotGeometryName = "myofiber_simplified"
)
# This extracts the segmented cells
cells <- annotGeometry(sfe_tissue, "myofiber_simplified") |>
st_geometry()
# We can also use ggplot and geom_sf to plot sf objects
q <- ggplot() +
geom_sf(data = cells, fill = NA) +
theme_void()
# Using `patchwork` to combine the plots
p | q
In lattice data analysis, a key concept is the spatial weight matrix, which models the spatial relationships between units in the lattice (i.e., spots or cells). Various methods exist for constructing this matrix, such as contiguity-based (direct neighbors), graph-based, or distance-based methods. (Getis 2009; Zuur, Ieno, and Smith 2007; Pebesma and Bivand 2023). The documentation of the package spdep gives an overview of the different methods.
For Visium, the most straightforward way is to take the direct neighbours of each spot. This is done using the function findVisiumGraph.
colGraph(sfe_tissue, "visium") <- findVisiumGraph(sfe_tissue)plotColGraph(sfe_tissue,
colGraphName = "visium",
colGeometryName = "spotPoly"
) + theme_void()
In an irregular lattice, the task of finding a spatial weight matrix is more complex, as different options exist. One option is to base the neighbourhood graph on neighbours that are in direct contact with each other (contiguous), as implemented in the poly2nb method.
annotGraph(sfe_tissue, "myofiber_poly2nb") <-
findSpatialNeighbors(sfe_tissue,
type = "myofiber_simplified",
MARGIN = 3,
method = "poly2nb", # wraps spdep function with same name
zero.policy = TRUE
)p1 <- plotAnnotGraph(sfe_tissue,
annotGraphName = "myofiber_poly2nb",
annotGeometryName = "myofiber_simplified"
) + theme_void()Alternatively, we could take the five nearest neighbours of each cell.
annotGraph(sfe_tissue, "knn5") <-
findSpatialNeighbors(sfe_tissue,
type = "myofiber_simplified",
MARGIN = 3, # to use the annotation geometry
method = "knearneigh", # wraps the spdep function with the same name
k = 5,
zero.policy = TRUE
)p2 <- plotAnnotGraph(sfe_tissue,
annotGraphName = "knn5",
annotGeometryName = "myofiber_simplified"
) + theme_void()As we can see below, the graphs look quite distinct. On the left side, in the contiguous neigbhbour graph (neighbours in direct contact) we can notice the formation of patches, while in the knn graph isolated patches are interconnected.
p1 + p2
Spatial autocorrelation measures the association between spatial units while recognizing that the spatial units are not independent measurements. These measures can be global (summarizing the entire field) or local (providing statistics for individual locations) (Pebesma and Bivand 2023).
Global methods give us an overview over the entire field-of-view and summarize the spatial autocorrelation metric to a single value. The metrics are a function of the weight matrix and the variables of interest. The variables of interest can be gene expression, intensity of a marker or the area of a cell. The global measures can be seen as a weighted average of the local metric, as explained below.
In general, a global spatial autocorrelation measure has the form of a double sum over all locations \(i,j\)
\[\sum_i \sum_j f(x_i,x_j) w_{ij}\]
where \(f(x_i,x_j)\) is the measure of association between features of interest and \(w_{ij}\) scales the relationship by a spatial weight as defined in the weight matrix \(W\) (Zuur, Ieno, and Smith 2007; Pebesma and Bivand 2023).
Moran’s I is the most prominent measure of spatial autocorrelation. The values are bounded by \(-1\) and \(1\). The expected value is close to \(0\) for large \(n\), the exact value is given by \(\mathbb{E}(I) = -1/(n-1)\). A value higher than the expected value indicates spatial auto-correlation. Negative values indicate negative auto-correlation. Spatial auto-correlation means that similar values tend to be found together in the tissue. In fact, Moran’s I can be interpreted as the Pearson correlation between the value at location \(i\) and the averages value of the neigbours of \(i\), (neighbours as defined in the weight matrix \(W\)) (Moran 1950).
In the first example we will calculate Moran’s I for the number of counts and genes measured in the Visium dataset. First, we have a look at the distribution by eye.
plotSpatialFeature(sfe_tissue,
features = c("nCounts", "nGenes"),
colGeometryName = "spotPoly",
swap_rownames = "symbol"
)
Based on these values we can calculate Moran’s I
calculateUnivariate(t(colData(sfe_tissue)[, c("nCounts", "nGenes")]),
type = "moran",
listw = colGraph(sfe_tissue, "visium")
)DataFrame with 2 rows and 2 columns
moran K
<numeric> <numeric>
nCounts 0.528705 3.00082
nGenes 0.384028 3.88036We can further use permutation testing to get a significance of our estimates.
sfe_tissue <- colDataUnivariate(sfe_tissue,
features = c("nCounts", "nGenes"),
colGraphName = "visium", nsim = 1000,
type = "moran.mc"
)
plotMoranMC(sfe_tissue, c("nCounts", "nGenes"),
linewidth = 2
)
Here we can see the Moran’s I values for the number of counts and genes that were identified in the Visium dataset and a permutation-based null distribution. As we can see, the Moran’s I values both indicate positive spatial autocorrelation and are highly significant. This means that regions with similar count and number of measured gene values tend to cluster together. This is an interesting finding and was observed in other spatial transcriptomic datasets, see (Bhuva et al. 2024).
Often a global measure is not enough. One number determining e.g. the spatial autocorrelation over an entire tissue slice might not be reflective of tissue heterogeneity. Therefore, local indicators of spatial associations have been developed (Pebesma and Bivand 2023).
Local Moran’s I provides a measure of spatial autocorrelation at each location, highlighting local clusters of similarity or dissimilarity (Anselin 1995). It is defined as:
\[I_i = \frac{x_i - \bar{x}}{\sum_{k=1}^n(x_k-\bar{x})^2/(n-1)} \sum_{j=1}^n w_{ij}(x_j - \bar{x})\]
where the index \(i\) refers to the location for which the measure is calculated. The interpretation is analogous to the global Moran’s I where a value of \(I_i\) higher than \(\mathbb{E}(I) = -1/(n-1)\) indicates spatial auto-correlation; smaller values indicate negative auto-correlation. It is important to note that, as for the global counterpart, the value of local Moran’s I could be a result from both the high or low end of the values. Here we will calculate the local Moran’s I value for the measurement of muscle fiber marker gene Myh2:
sfe_tissue <- runUnivariate(sfe_tissue,
type = "localmoran",
features = "Myh2",
colGraphName = "visium",
swap_rownames = "symbol"
)
# plot the expression values
pExp <- plotSpatialFeature(sfe_tissue,
features = c("Myh2"), colGeometryName = "spotPoly",
swap_rownames = "symbol"
)
# plot the local Moran's I values
pLi <- plotLocalResult(sfe_tissue, "localmoran",
features = c("Myh2"),
colGeometryName = "spotPoly", swap_rownames = "symbol",
divergent = TRUE, diverge_center = 0
) + labs(fill = "li(Myh2)") # specify legend
# plot the local Moran's I p-values
pPval <- plotLocalResult(sfe_tissue, "localmoran",
features = c("Myh2"), "-log10p_adj",
colGeometryName = "spotPoly", swap_rownames = "symbol",
divergent = TRUE, diverge_center = -log10(0.05)
) + labs(fill = "-log10(p.adj)") # specify legend
# please note that the
pExp + pLi + pPval
In the local version of Moran’s I, the interpretation is the same as the global version. When interpreting local autocorrelation measures, it is important to consider both the effect size estimates and the significance level. Since the significance level is calculated for each spot separately, it is recommended to adjust for multiple testing. By default, runUnivariate uses the Benjamini & Hochberg correction. The local Moran’s I statistics reveal locations in the tissue that have similar values to their neighbours (c.f., the lower part of the tissue).
There exists a bivariate version of Moran’s I as well. The package spatialDM (Li et al. 2023) uses an adapted version of bivariate Moran’s I to identify ligand-receptor pairs.
For two continous observations the global bivariate Moran’s I is defined as (Wartenberg 1985; Bivand 2022)
\[I_B = \frac{\Sigma_i(\Sigma_j{w_{ij}b_j\times a_i})}{\Sigma_i{b_i^2}}\]
where \(a_i\) and \(b_i\) are the two variables of interest and \(w_{ij}\) is the value of the spatial weights matrix for positions \(i\) and \(j\). It is a measure of the correlation of the variables \(a\) with the the average of the neighboring values for variable \(b\) (also called the spatial lag of \(b\)).
There exists a local version that we will explore here. Note that the results are not symmetric, but very similar to each other.
sfe_tissue <- runBivariate(sfe_tissue, "localmoran_bv", # wraps the method from spdep
c("Myh1", "Myh2"),
swap_rownames = "symbol", nsim = 1000
)
# this command gives all results of localmoran_bv
localResultFeatures(sfe_tissue, "localmoran_bv")[1] "Myh1__Myh1" "Myh2__Myh1" "Myh1__Myh2" "Myh2__Myh2"plotLocalResult(sfe_tissue, "localmoran_bv", c("Myh1__Myh2", "Myh2__Myh1"),
colGeometryName = "spotPoly", divergent = TRUE, diverge_center = 0
)
Please note that the result might overestimate the spatial autocorrelation of the variables due to the inherent (non-spatial) correlation of \(x\) and \(y\) (Bivand 2022).
Let’s compare the impact of different spatial weight matrices on local autocorrelation analysis. In this Visium dataset we do not have gene expression information for each cell because of the resolution of the spots. We will instead calculate the autocorrelation measures on the area of the segmented cells. This could be helpful when looking at local cell densities in a tissue.
First, we base the spatial weight matrix on contiguous neighbours.
sfe_tissue <- annotGeometryUnivariate(sfe_tissue, "localmoran", "area",
annotGeometryName = "myofiber_simplified",
annotGraphName = "myofiber_poly2nb",
include_self = FALSE, zero.policy = TRUE,
name = "myofiber_poly2nb"
)
pPoly <- plotLocalResult(sfe_tissue, "myofiber_poly2nb", "area",
annotGeometryName = "myofiber_simplified",
annotGraphName = "myofiber_poly2nb",
divergent = TRUE, diverge_center = 0
) + labs(title = "Local Moran's I (li)\nContiguous Neighbours",
fill = "li(area)")As an alternative, we also base the spatial weight matrix on the 5 nearest neighbours.
sfe_tissue <- annotGeometryUnivariate(sfe_tissue, "localmoran", "area",
annotGeometryName = "myofiber_simplified",
annotGraphName = "knn5",
include_self = FALSE, zero.policy = TRUE,
name = "knn5"
)
pKnn5 <- plotLocalResult(sfe_tissue, "knn5", "area",
annotGeometryName = "myofiber_simplified",
annotGraphName = "knn5",
divergent = TRUE, diverge_center = 0
) + labs(title = "Local Moran's I (li)\n5 Nearest Neighbours",
fill = "li(area)")pPoly + pKnn5
What happens if we base the spatial weight matrix on 10 nearest neighbours?
annotGraph(sfe_tissue, "knn10") <-
findSpatialNeighbors(sfe_tissue,
type = "myofiber_simplified",
MARGIN = 3, # to use the annotation geometry
method = "knearneigh", # wraps the spdep function with the same name
k = 10,
zero.policy = TRUE
)
sfe_tissue <- annotGeometryUnivariate(sfe_tissue, "localmoran", "area",
annotGeometryName = "myofiber_simplified",
annotGraphName = "knn10",
include_self = FALSE, zero.policy = TRUE,
name = "knn10"
)
pKnn10 <- plotLocalResult(sfe_tissue, "knn10", "area",
annotGeometryName = "myofiber_simplified",
annotGraphName = "knn10",
divergent = TRUE, diverge_center = 0
) + labs(title = "Local Moran's I (li)\n10 Nearest Neighbours",
fill = "li(area)")pPoly + pKnn5 + pKnn10
The overall pattern of spatial autocorrelation is similar across the different weight matrices. However, locally there are some differences, as well as in the scale of the autocorrelation (note that the same color in the different plots does not correspond to the same value between the plots). There is a trend of smoothing of the value, the larger the neighbourhood in consideration.
sessionInfo()R version 4.3.1 (2023-06-16)
Platform: aarch64-apple-darwin20 (64-bit)
Running under: macOS Sonoma 14.5
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRlapack.dylib; LAPACK version 3.11.0
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
time zone: Europe/Zurich
tzcode source: internal
attached base packages:
[1] stats4 stats graphics grDevices utils datasets methods
[8] base
other attached packages:
[1] scater_1.28.0 scuttle_1.10.3
[3] SingleCellExperiment_1.22.0 SummarizedExperiment_1.30.2
[5] Biobase_2.60.0 GenomicRanges_1.52.1
[7] GenomeInfoDb_1.36.4 IRanges_2.34.1
[9] S4Vectors_0.38.2 BiocGenerics_0.46.0
[11] MatrixGenerics_1.12.3 matrixStats_1.0.0
[13] magrittr_2.0.3 tidyr_1.3.0
[15] stringr_1.5.0 spdep_1.2-8
[17] sf_1.0-16 spData_2.3.0
[19] SFEData_1.2.0 SpatialFeatureExperiment_1.2.3
[21] Voyager_1.2.7 patchwork_1.2.0
[23] ggplot2_3.5.1 dplyr_1.1.3
loaded via a namespace (and not attached):
[1] RColorBrewer_1.1-3 rstudioapi_0.15.0
[3] jsonlite_1.8.7 wk_0.8.0
[5] ggbeeswarm_0.7.2 magick_2.8.0
[7] farver_2.1.1 rmarkdown_2.25
[9] zlibbioc_1.46.0 vctrs_0.6.4
[11] memoise_2.0.1 DelayedMatrixStats_1.22.6
[13] RCurl_1.98-1.12 terra_1.7-55
[15] htmltools_0.5.6.1 S4Arrays_1.0.6
[17] AnnotationHub_3.8.0 curl_5.1.0
[19] BiocNeighbors_1.18.0 Rhdf5lib_1.22.1
[21] s2_1.1.4 rhdf5_2.44.0
[23] KernSmooth_2.23-21 htmlwidgets_1.6.2
[25] cachem_1.0.8 igraph_1.5.1
[27] mime_0.12 lifecycle_1.0.3
[29] pkgconfig_2.0.3 rsvd_1.0.5
[31] Matrix_1.5-4.1 R6_2.5.1
[33] fastmap_1.1.1 GenomeInfoDbData_1.2.10
[35] shiny_1.7.5.1 digest_0.6.33
[37] colorspace_2.1-0 ggnewscale_0.4.9
[39] AnnotationDbi_1.62.2 irlba_2.3.5.1
[41] dqrng_0.3.1 RSpectra_0.16-1
[43] ExperimentHub_2.8.1 RSQLite_2.3.1
[45] beachmat_2.16.0 labeling_0.4.3
[47] filelock_1.0.2 fansi_1.0.5
[49] httr_1.4.7 abind_1.4-5
[51] compiler_4.3.1 proxy_0.4-27
[53] bit64_4.0.5 withr_2.5.1
[55] BiocParallel_1.34.2 viridis_0.6.4
[57] DBI_1.1.3 HDF5Array_1.28.1
[59] R.utils_2.12.2 rappdirs_0.3.3
[61] DelayedArray_0.26.7 rjson_0.2.21
[63] classInt_0.4-10 bluster_1.10.0
[65] tools_4.3.1 units_0.8-4
[67] vipor_0.4.5 beeswarm_0.4.0
[69] interactiveDisplayBase_1.38.0 httpuv_1.6.11
[71] R.oo_1.25.0 glue_1.6.2
[73] dbscan_1.1-11 rhdf5filters_1.12.1
[75] promises_1.2.1 grid_4.3.1
[77] cluster_2.1.4 generics_0.1.3
[79] gtable_0.3.4 R.methodsS3_1.8.2
[81] class_7.3-22 ScaledMatrix_1.8.1
[83] BiocSingular_1.16.0 sp_2.1-1
[85] utf8_1.2.3 XVector_0.40.0
[87] ggrepel_0.9.4 BiocVersion_3.17.1
[89] pillar_1.9.0 limma_3.56.2
[91] later_1.3.1 BiocFileCache_2.8.0
[93] lattice_0.21-8 bit_4.0.5
[95] deldir_1.0-9 tidyselect_1.2.0
[97] locfit_1.5-9.8 Biostrings_2.68.1
[99] knitr_1.44 gridExtra_2.3
[101] edgeR_3.42.4 xfun_0.40
[103] DropletUtils_1.20.0 stringi_1.7.12
[105] yaml_2.3.7 boot_1.3-28.1
[107] evaluate_0.22 codetools_0.2-19
[109] tibble_3.2.1 BiocManager_1.30.22
[111] cli_3.6.1 xtable_1.8-4
[113] munsell_0.5.0 Rcpp_1.0.11
[115] dbplyr_2.3.4 png_0.1-8
[117] parallel_4.3.1 ellipsis_0.3.2
[119] blob_1.2.4 sparseMatrixStats_1.12.2
[121] bitops_1.0-7 SpatialExperiment_1.10.0
[123] viridisLite_0.4.2 scales_1.3.0
[125] e1071_1.7-13 purrr_1.0.2
[127] crayon_1.5.2 BiocStyle_2.28.1
[129] scico_1.5.0 rlang_1.1.1
[131] KEGGREST_1.40.1 This tutorial is partially based on the following Voyager vignette.
©2024 The pasta authors. Content is published under Creative Commons CC-BY-4.0 License for the text and GPL-3 License for any code.